continuous mapping theorem for function of two variables from Hilbert space

Question

Assume we are in $\ell_{2}$ space and we have two sequences $X_{n}$ and $Y_{n}$ with values from $\ell_{2}$ such that $||X_{n}||_{2} < c_{1}$ and $||Y_{n}||_{2} < c_{2}$ and $$ X_{n} \stackrel{a.s.}{\to} a $$ and $$ Y_{n} \stackrel{a.s.}{\to} b $$

Is this true that

$$ <X_{n},Y_{n} > \stackrel{a.s.}{\to} <a,b>? $$

Would a.s. convergence hold for any continuous function $\ell_{2} \times \ell_{2} \to \mathbb{R}$? What is continuous mapping theorem in this case?

Answer 1

Yes. Indeed, if Xn and Yn are sequences converging to X and Y (respectively) almost surely, then (Xn,Yn) converges almost surely (in the direct sum of l2 with itself) to (X,Y). Since the inner product is continuous, this proves the claim.

Answer 2

The map from $\ell^2 \times \ell^2 \to \mathbb{C}$ defined by $(x, y) \mapsto \langle x, y \rangle$ is continuous because of the estimate $|\langle x, y \rangle| \leq \|x\|\|y\|$, known as the Cauchy-Schwarz inequality.